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Could you please try to find the intersections points of this two functions: $ y = x $ and $y=(a+x) \ {\rm e}^ {-2(a+x)} $ with $x\geq0$

Thanks for your help

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    $\begingroup$ Hi Ivan, Welcome to MSE! Can you please post the code you have tried. $\endgroup$ Sep 26, 2019 at 14:50
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    $\begingroup$ Try the Lambert function en.wikipedia.org/wiki/Lambert_W_function $\endgroup$
    – yarchik
    Sep 26, 2019 at 14:55
  • $\begingroup$ see also ProductLog $\endgroup$
    – kglr
    Sep 26, 2019 at 15:04
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    $\begingroup$ The generalization of the Lambert function by Mező real.mtak.hu/39795/1/… is relevant $\endgroup$
    – yarchik
    Sep 26, 2019 at 15:13
  • $\begingroup$ @yarchik Looks amazing. $\endgroup$ Sep 27, 2019 at 11:05

2 Answers 2

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To visualize the pairs of real numbers {x, a} that satisfy the equation,

you can use ContourPlot:

ContourPlot[(a + x) E^(-2 (a + x)) == x, {x, -2, 1}, {a, 0, 5}]

enter image description here

or RegionPlot with options MeshFunctions + Mesh:

RegionPlot[True, {x, -2, 1}, {a, 0, 5}, 
 BoundaryStyle -> None, 
 MeshFunctions -> {(#2 + #) E^(-2 (#2 + #)) - # &}, 
 Mesh -> {{0}}, 
 MeshStyle -> Directive[Thick, Red], 
 PlotStyle -> None, 
 PlotPoints -> 100]

enter image description here

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Looks like you'll need some constraints on a.

a = -1;
f[x_] := (a + x) E^(-2 (a + x))
Plot[{x, f[x]}, {x, 0.001, 4}, PlotRange -> {Automatic, {-2, 1}}]

enter image description here

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